Question

101-2019-3-b (1).pdf-Adobe Acrobat Reader DC Eile Edit iew Window Help Home Tools 101-2019-3-b (1) Sign In x Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y, x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V -> V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace Uo or U-V, if and only if, for every linear transformation V 4V on V, U is an invariant subspace of f Hint: Extend a basis of U to a basis of V. (c) Show that if λί. Α2, , λk are distinct eigenvalues of a matrix A and xi E EM [A] ( 1,2,.. ,n) then x1,X2, . .. ,Xn^ is a linearly independent set of vectors Hint: Assume not and use the fact: given any set of finitely many dependent vectors there exists a smallest subset which is linearly dependent. (Why? Convince yourself with a separate proof of this statement.) (d) If V ^V be a linear transformation with dim Imf k then f has at most k +1 distinot eigenvalues Hint: Use (c (e) Show that a matrix is singular if and only if 0 is one of its eigenvalues (f) Suppose A is an invertible square matrix. Show that the characteristic polynomial of A-1 is a scalar multiple of the characteristic polynomial of A What is the scalar multiple equal to?

Answer 1

a)tェ.tpt b--topt).2.14n) stamdard b tel λ be eigenvalue A 0 Jon

Conidex